Radian

An angle at the centre of a circle cuts off an arc. Its size in radians is the length of that arc divided by the radius, $\theta = s/r$: the number of radii the arc is long. For a fixed angle the arc grows in proportion to the radius, so this ratio does not depend on how large the circle is. One radian is the angle whose arc is exactly as long as the radius, $180^\circ/\pi \approx 57.2958^\circ$; a full turn is $2\pi$ radians, because the circumference is $2\pi r$. Measured this way, an arc is $s = r\theta$ and a sector has area $\tfrac12 r^2\theta$.

In the picture the radius OA is the ruler. Measure with r lays it along the arc piece by piece, bending each piece onto the circle without changing its length. The arc is ticked at every radius of length. Six whole radii fit around the circle, and $2\pi - 6 \approx 0.2832$ of a radius is left over. On the concentric circles of radius $r/3$ and $2r/3$, the ticks are spaced by each circle's own radius, and they fall on the same rays: the angle does not depend on the size of the circle.

Why calculus uses radians: in the unit circle the triangle OAP lies inside the sector OAP, which lies inside the triangle OAT, where T is on the tangent at A. Their areas are $\tfrac12\sin\theta$, $\tfrac12\theta$ and $\tfrac12\tan\theta$; dividing by $\tfrac12\sin\theta$ and taking reciprocals gives the squeeze above. As $\theta \to 0$, $\cos\theta \to 1$, so $\sin\theta/\theta \to 1$. The ratio is even, so the limit from the left is the same. That limit is the slope of sin at 0, the first step to $(\sin x)' = \cos x$.

Radians come in through the sector's area, $\tfrac12\theta$. Measure the angle in degrees, $x^\circ$, and the sector's area becomes $(\pi/360)\,x$. As $x \to 0$, the same squeeze gives $\sin x^\circ/x \to \pi/180 \approx 0.017453$, so in degrees sin has slope π/180 at 0, not 1.